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PT -Symmetric Models in Classical and Quantum Mechanics
... oscillator and exponential growth in the other. The phase transition is observed by variation of the relevant parameters. This classical situation is analogous to the phase transition between real and complex eigenvalues in a quantum system defined by a PT -symmetric Hamiltonian. To construct a viab ...
... oscillator and exponential growth in the other. The phase transition is observed by variation of the relevant parameters. This classical situation is analogous to the phase transition between real and complex eigenvalues in a quantum system defined by a PT -symmetric Hamiltonian. To construct a viab ...
An introduction to rigorous formulations of quantum field theory
... begin,“The mathematical data of a quantum field theory consists of a Hilbert space H, a self-adjoint Hamiltonian operator H acting on H,” and so on. Given a particular quantum field theory with some spacetime and Lagrangian, the challenge is then to construct this theory within the language of the a ...
... begin,“The mathematical data of a quantum field theory consists of a Hilbert space H, a self-adjoint Hamiltonian operator H acting on H,” and so on. Given a particular quantum field theory with some spacetime and Lagrangian, the challenge is then to construct this theory within the language of the a ...
A note on closedness of algebraic sum of sets
... such that the sets x + A and B are disjoint, and hence (by what we have just proved above) the set (x + A) − B is closed. But then the set [(x + A) − B] − x = A − B is closed too. Finally from the equality A + B = A − (−B) we obtain that the sum A + B is closed for any two closed convex and bounded ...
... such that the sets x + A and B are disjoint, and hence (by what we have just proved above) the set (x + A) − B is closed. But then the set [(x + A) − B] − x = A − B is closed too. Finally from the equality A + B = A − (−B) we obtain that the sum A + B is closed for any two closed convex and bounded ...
Classifying spaces and spectral sequences
... Let G be a topological group. It can be identified with a topological category with ob(G)== point, mor(G)=G. Its semi-simplicial space NG is given by NG^=G^=:Gx. . . X G (k times). The space BG if often a classifying space for G in the usual sense, as one can see as follows. Consider the category G ...
... Let G be a topological group. It can be identified with a topological category with ob(G)== point, mor(G)=G. Its semi-simplicial space NG is given by NG^=G^=:Gx. . . X G (k times). The space BG if often a classifying space for G in the usual sense, as one can see as follows. Consider the category G ...
APERIODIC ORDER – LECTURE 6 SUMMARY 1. Elements of
... Prove that this sequence is positive definite. Definition 1.12. Let f ∈ L2 (X, µ). The spectral type %f of f is the finite Borel measure on the circle T such that %b(n) = hUTn f, f i for n ∈ Z. Exercise. Let f be an eigenfunction corresponding to an eigenvalue λ of unit norm: kf k2 = 1. Prove that % ...
... Prove that this sequence is positive definite. Definition 1.12. Let f ∈ L2 (X, µ). The spectral type %f of f is the finite Borel measure on the circle T such that %b(n) = hUTn f, f i for n ∈ Z. Exercise. Let f be an eigenfunction corresponding to an eigenvalue λ of unit norm: kf k2 = 1. Prove that % ...
coherent states in quantum mechanics
... In classical physics the properties of a certain system can be described using its position x and mass m. With these variables it is possible to determine the velocity v(=dx/dt), the momentum p(=mv) and any other dynamical variable of interest. Quantum mechanics describes the time evolution of physi ...
... In classical physics the properties of a certain system can be described using its position x and mass m. With these variables it is possible to determine the velocity v(=dx/dt), the momentum p(=mv) and any other dynamical variable of interest. Quantum mechanics describes the time evolution of physi ...
Orthogonal Polynomials
... space of all functions, the orthogonal polynomials p0 , . . . pk constitute an “orthogonal basis” for the subspace of polynomial functions of degree no more than k. The least-squares approximation of a function f by polynomials in this subspace is then its orthogonal projection onto the subspace. Th ...
... space of all functions, the orthogonal polynomials p0 , . . . pk constitute an “orthogonal basis” for the subspace of polynomial functions of degree no more than k. The least-squares approximation of a function f by polynomials in this subspace is then its orthogonal projection onto the subspace. Th ...
QUANTUM FIELD THEORY ON CURVED
... where F and hµν are only functions of x1 , . . . , xd−1 . It is clear from (1.4) that the natural time-translation and time-reflection maps are isometries for all points in the neighborhood where these coordinates are defined. 1.2. Analytic continuation. The Euclidean approach to quantum field theor ...
... where F and hµν are only functions of x1 , . . . , xd−1 . It is clear from (1.4) that the natural time-translation and time-reflection maps are isometries for all points in the neighborhood where these coordinates are defined. 1.2. Analytic continuation. The Euclidean approach to quantum field theor ...
Morse Theory is a part pf differential geometry, concerned with
... forms. We will also define the Hodge operator, which will be used in the proof of the Morse Inequalities. If we choose a Riemannian metric on the manifold M, denoted gm , this implies that for all m M , we have an inner product gm (, ) on the tangent space T ( M ). For smooth vector fields on ...
... forms. We will also define the Hodge operator, which will be used in the proof of the Morse Inequalities. If we choose a Riemannian metric on the manifold M, denoted gm , this implies that for all m M , we have an inner product gm (, ) on the tangent space T ( M ). For smooth vector fields on ...
M10/17
... equivalent if there exists a unitary operator U : H → K such that U ψ = φ and U E(A)U ∗ = F(A) for all A ∈ A. For example, if (H, E, ψ) is an operator representation for D and α ∈ C with |α| = 1, then (H, E, αψ) is an equivalent operator representation for D. In this case, the unitary operator is U ...
... equivalent if there exists a unitary operator U : H → K such that U ψ = φ and U E(A)U ∗ = F(A) for all A ∈ A. For example, if (H, E, ψ) is an operator representation for D and α ∈ C with |α| = 1, then (H, E, αψ) is an equivalent operator representation for D. In this case, the unitary operator is U ...
Integral and differential structures for quantum field theory
... for any v, w ∈ D, and any n ∈ IN. In other words, the number (v, φn (f )w) is finite for any n ∈ IN. Corollary 2.2. For states ωx (·) ≡ (x, ·x) with x ∈ D, field operators in QFT enjoy the property of having all moments finite. We remind that this feature is a starting point for an analysis of appli ...
... for any v, w ∈ D, and any n ∈ IN. In other words, the number (v, φn (f )w) is finite for any n ∈ IN. Corollary 2.2. For states ωx (·) ≡ (x, ·x) with x ∈ D, field operators in QFT enjoy the property of having all moments finite. We remind that this feature is a starting point for an analysis of appli ...
Lecture 22 Relevant sections in text: §3.1, 3.2 Rotations in quantum mechanics
... where ω12 is a real number, which may depend upon the choice of rotations R1 and R2 , as its notation suggests. This phase freedom is allowed since the state vector D(R1 R2 )|ψi cannot be physically distinguished from eiω12 D(R1 R2 )|ψi. If we succeed in constructing this family of unitary operators ...
... where ω12 is a real number, which may depend upon the choice of rotations R1 and R2 , as its notation suggests. This phase freedom is allowed since the state vector D(R1 R2 )|ψi cannot be physically distinguished from eiω12 D(R1 R2 )|ψi. If we succeed in constructing this family of unitary operators ...
Fields and vector spaces
... where m and n are fixed but arbitrary natural numbers. This set (and space) is denoted Mm,n (F ) or F m×n ; in the square case m = n we may just write Mn (F ). If A and B are m × n matrices with entries from F and 1 ≤ j ≤ m, 1 ≤ k ≤ n, then the (j, k)-entry of A + B is aj,k + bj,k where aj,k is the ...
... where m and n are fixed but arbitrary natural numbers. This set (and space) is denoted Mm,n (F ) or F m×n ; in the square case m = n we may just write Mn (F ). If A and B are m × n matrices with entries from F and 1 ≤ j ≤ m, 1 ≤ k ≤ n, then the (j, k)-entry of A + B is aj,k + bj,k where aj,k is the ...
Trace Ideal Criteria for Hankel Operators and Commutators
... studying commutators is that they are, in various senses, building blocks from which other more complicated operators are constructed. Because of this, \-ve can use Proposition 6 as a starting point and obtain trace class criteria for other operators. We now give several examples. \Ve will say of tw ...
... studying commutators is that they are, in various senses, building blocks from which other more complicated operators are constructed. Because of this, \-ve can use Proposition 6 as a starting point and obtain trace class criteria for other operators. We now give several examples. \Ve will say of tw ...
Another property of the Sorgenfrey line
... open cover of X x Y which is closed under finite unions, then 7i’ has a a-locally finite refinement. Let W be such a cover. Let F(n)> be a spectral 1-network for Y, as in (2.4). For each n ~ 1 and for each ntuple (03B11, ···, an) of elements of A, let 4Y(ai , ... , 03B1n) = {R~X: R is open and ...
... open cover of X x Y which is closed under finite unions, then 7i’ has a a-locally finite refinement. Let W be such a cover. Let F(n)> be a spectral 1-network for Y, as in (2.4). For each n ~ 1 and for each ntuple (03B11, ···, an) of elements of A, let 4Y(ai , ... , 03B1n) = {R~X: R is open and ...
Quantum Field Theory on Curved Backgrounds. II
... Let D = d/dt denote the canonical unit vector field on R. Let G be a real Lie group with algebra g, and let X ∈ g. The map tD → tX(t ∈ R) is a homomorphism of Lie(R) → g, so by the Lemma there is a unique analytic homomorphism ξX : R → G such that d ξX (D) = X. Conversely, if η is an analytic homomo ...
... Let D = d/dt denote the canonical unit vector field on R. Let G be a real Lie group with algebra g, and let X ∈ g. The map tD → tX(t ∈ R) is a homomorphism of Lie(R) → g, so by the Lemma there is a unique analytic homomorphism ξX : R → G such that d ξX (D) = X. Conversely, if η is an analytic homomo ...
Geometric Quantization - Texas Christian University
... The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical example of this is the cotangent bundle of a manifold. The manifold is the configuration space (ie set of positions), and the tangent bund ...
... The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical example of this is the cotangent bundle of a manifold. The manifold is the configuration space (ie set of positions), and the tangent bund ...
... 3. Existence and uniqueness. Evolution problems of the form (2.5) have been considered by many writers, and we refer to the recent work of M. Crandall and A. Pazy [10] and J. Dorroh [12] for references in this direction. In particular, a sufficient condition for uniqueness of solutions of Cauchy pro ...
Answer - UIUC Math
... L(y) T implies that L(x) + L(y) T and L(x + y) T . Thus if we have two elelments x, y V we know that L(x + y) T . In other words x + y L−1 (T ) for any x and y in L−1 (T ) (by the given definition). This completes the proof. ...
... L(y) T implies that L(x) + L(y) T and L(x + y) T . Thus if we have two elelments x, y V we know that L(x + y) T . In other words x + y L−1 (T ) for any x and y in L−1 (T ) (by the given definition). This completes the proof. ...
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of ""dropping the altitude"" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.